How does one derive the lens maker's formula for thick lens? The well-known equation for thin lens is:
$$\frac{1}{f}=\left(\frac{n_L}{n_m}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$
But there's a more appropriate equation that includes the thickness of the lens, which is:
$$\frac{1}{f}=\left(\frac{n_L}{n_m}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}+\frac{(\frac{n_L}{n_m}-1)d}{n_LR_1R_2}\right)$$
However, I can not find any derivation of it online. As far as I managed to find is the derivation of the following:
$$\frac{1}{f}=\left(\frac{n_L}{n_m}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}+\frac{n_Ld}{s_{i1}(s_{i1}-d)}\right)$$
which can be found at page 167 (Equation 5.14) of Optics by Eugene Hecht:
https://edisciplinas.usp.br/pluginfile.php/5054148/mod_resource/content/1/Hecht-optics-5ed.pdf
 A: Here is the answer I came up with after thinking about my question again.
Consider the derivation of the common lens equation (See below the picture of the linked derivation). Note that one of the steps is to consider the height at which the beam of light enters and exits the lens to be the same. To account for the thickness of the lens, one needs to consider the difference in these heights.

Note that
$$\tan(\theta_7)\approx\frac{h_1-h_2}{d}$$
Then, using trigonometric approximations and the equations in the derivation,
$$\theta_7\approx\frac{h_1-h_2}{d}$$
$$\theta_1-\theta_2\approx\frac{h_1-h_2}{d}$$
$$\frac{h_1}{R_1}-\frac{h_1}{nR_1}\approx\frac{h_1-h_2}{d}$$
Therefore,
$$\frac{h_1}{h_2}=\frac{1}{1-\frac{d}{nR_1}(n-1)}\approx1+\frac{d(n-1)}{nR_1}$$
Substituting this equation at step (4) of the derivation gives
$$\frac{1}{f}=(n-1)\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{d(n-1)}{nR_1^2}\right)$$
However, I recommend not approximating $\frac{h_1}{h_2}$. If you approximate it, as done in the above formula, the result will be more accurate than the common lens equation, but worse than the original thick lens equation. Using it without approximation (which is better than the common and original thick lens formula) the result is
$$\frac{1}{f}=(n-1)\left(\frac{1}{R_1-\frac{d(n-1)}{n}}+\frac{1}{R_2}\right)$$
Also, note that:

*

*The difference in sign of the $\frac{1}{R_2}$ term seems to be due to sign convention

*The definition of $f$ here is the horizontal distance between the point which light exits the lens (which is approximately the right end of the lens) and where it is focused

*$R_1$ is the radius of the side the light enters the lens.

A: I was looking into this myself this week. A very thorough derivation can be found here: http://www.learnoptics.com/Optics%20Chapters/opticsintrochap1.pdf. Unfortunately, you will likely have to read all until page 46. Luckily, it closely mirrors the Hecht book and is very informative. Note the ray transfer matrices follow a different convention from Hecht in which the terms of the ABCD matrices are rearranged.
Alternately, a simpler derivation would be by using ray transfer matrices as shown here:https://www.youtube.com/watch?v=nNFhsmVlyeE
A: A spherical thick lens, of index of refraction na, immersed between two different optical media, with index of refraction n1 and n2, consist in two refractive spherical surfaces of radios R1a and R2a, separated a distance da over the optical axis, see Figure bellow:

In the figure,  a generic optical ray (in brown color) is propagating in the first media with index of refraction n1 with a alpha1 angle with respect to the optical axis. Then the ray  is refracted by the first surface of the lens, at a height y1. After that the ray propagates inside the lens with refraction index na (a distance da measured over the optical axis) and is refracted by the second surface, leaving the lens with an angle of alpha2 at a height y2 with respect to the optical axis, following further propagation into the n2 medium. The mathematical relation between angles and heights in the two surfaces can be obtained using the optical matrix formalism in the following way:

Performing the matrix product:

The expression corresponding to the matrix element a12 is:

Which is the focal distance for the thick lens. Making n1=n2=1, it is obtained the expression for the focal distance (or effective focal distance, as is called by Hetch in "Optics"), for a thick lens inmersed in air.
See http://pubs.sciepub.com/ijp/9/6/1/ for more details.
